标题： 关于伯特尔森-格罗莫夫动力莫尔斯熵 显示英文标题

标题： On Bertelson-Gromov Dynamical Morse Entropy

摘要： 在本文主要的说明性论文中，我们给出了M. Bertelson和M. Gromov在关于动力学莫尔斯熵的论文中包含的几个结果的详细证明。这是对该著作中提出的思想的介绍。假设$M$是一个有限维的紧致定向连通$C^\infty$流形。假设$f_0 :M \to [0,1]$是一个满射的莫尔斯函数。 对于给定的自然数$n$，考虑集合$M^n$并对于$x=(x_0,x_1,...,x_{n-1}) \in M^n$，记$ f_n (x) = \frac{1}{n} \, \sum_{j=0}^{n-1} f_0 (x_j).$。动力学莫尔斯熵描述了在固定区间$I\subset [0,1]$时，当$n \to \infty$时，$f_n$在$I$中的临界点数量的渐近增长。 与贝蒂数熵相关的部分不需要微分结构。 可以使用这种工具描述统计力学中$XY$模型定义的势能的普遍性质。 显示英文摘要

摘要： In this mainly expository paper we present a detailed proof of several results contained in a paper by M. Bertelson and M. Gromov on Dynamical Morse Entropy. This is an introduction to the ideas presented in that work. Suppose $M$ is compact oriented connected $C^\infty$ manifold of finite dimension. Assume that $f_0 :M \to [0,1]$ is a surjective Morse function. For a given natural number $n$, consider the set $M^n$ and for $x=(x_0,x_1,...,x_{n-1}) \in M^n$, denote $ f_n (x) = \frac{1}{n} \, \sum_{j=0}^{n-1} f_0 (x_j).$ The Dynamical Morse Entropy describes for a fixed interval $I\subset [0,1]$ the asymptotic growth of the number of critical points of $f_n$ in $I$, when $n \to \infty$. The part related to the Betti number entropy does not requires the differentiable structure. One can describe generic properties of potentials defined in the $XY$ model of Statistical Mechanics with this machinery.